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% This document contains the chapter about other transmission lines.
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% Copyright (C) 2006, 2007, 2009, 2011 Stefan Jahn <stefan@lkcc.org>
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\chapter{Other types of transmission lines}
%\addcontentsline{toc}{chapter}{Other types of transmission lines}

\section{Coaxial cable}
%\addcontentsline{toc}{section}{Coaxial cable}

\begin{figure}[ht]
\begin{center}
\psfrag{er}{$\mathrm{\varepsilon_r}$}
\includegraphics[width=0.35\linewidth]{coax}
\end{center}
\caption{coaxial line}
\label{fig:coax}
\end{figure}
\FloatBarrier

\subsection{Characteristic impedance}
%\addcontentsline{toc}{subsection}{Characteristic impedance}

The characteristic impedance of a coaxial line can be calculated as follows:
\begin{equation}
Z_L = \dfrac{Z_{F0}}{2\pi\cdot\sqrt{\varepsilon_r}}\cdot\ln{\left(\dfrac{D}{d}\right)}
\end{equation}

\subsection{Losses}
%\addcontentsline{toc}{subsection}{Losses}

Overall losses in a coaxial cable consist of dielectric and conductor
losses.  The dielectric losses compute as follows:
\begin{equation}
\alpha_d = \dfrac{\pi}{c_0}\cdot f\cdot \sqrt{\varepsilon_r} \cdot \tan{\delta}
\end{equation}

The conductor (i.e. ohmic) losses are specified by
\begin{equation}
\alpha_c = \sqrt{\varepsilon_r} \cdot\left(\dfrac{\dfrac{1}{D} + \dfrac{1}{d}}{\ln{\left(\dfrac{D}{d}\right)}}\right)\cdot\dfrac{R_S}{Z_{F0}}
\end{equation}

with $R_S$ denoting the sheet resistance of the conductor material,
i.e. the skin resistance
\begin{equation}
R_S = \sqrt{\pi\cdot f\cdot \mu_r \cdot \mu_o \cdot \rho}
\end{equation}

\subsection{Cutoff frequencies}
%\addcontentsline{toc}{subsection}{Cutoff frequencies}

In normal operation a signal wave passes through the coaxial line as a
TEM wave with no electrical or magnetic field component in the
direction of propagation.  Beyond a certain cutoff frequency
additional (unwanted) higher order modes are excited.
\begin{align}
f_{TE} &\approx \dfrac{2\cdot c_0}{\pi\cdot\left(D + d\right)\cdot\sqrt{\varepsilon_r}}
\;\;\;\;\rightarrow\;\;\;\; \textrm{TE(1,1) mode}\\
f_{TM} &\approx \dfrac{c_0}{\left(D - d\right)\cdot\sqrt{\varepsilon_r}}
\;\;\;\;\rightarrow\;\;\;\; \textrm{TM(n,1) mode}
\end{align}

\section{Rectangular waveguide}
\label{sec:rectWG}

In most of practical applications, the rectangular waveguides are designed to propagate the dominant mode only. This allows the designer to use conventional transmission line equations and avoid power leaking to higher order modes. In this sense, the Qucs rectangular waveguide component was implemented to extract the transmission line parameters from the $TE_{10}$ mode (dominant mode) and generate the quadripole parameters. For this reason, the simulation data is valid only for those frequencies between the $TE_{10}$ and $TM_{01}$ cutoff, where only $TE_{10}$ propagation is guaranteed.


\begin{figure}[H]
\begin{center}
\psfrag{er}{$\mathrm{\varepsilon_r}$}
\includegraphics[width=0.5\linewidth]{rectwaveguide}
\end{center}
\caption{Rectangular waveguide of width \textit{a}, height \textit{b} and length \textit{L}}
\label{fig:rectwg}
\end{figure}


\noindent As detailed in \ref{sec:tl_yparameter}, the quadripole parameters of an ideal transmission line depends on the propagation constant ($\gamma$), the charateristic impedance of the line ($Z_0$) and its length (L). \\ 

The propagation constant is defined as $\gamma = \alpha + j \cdot \beta$, where $\alpha$ is the attenuation constant and $\beta$ is the phase constant. The attenuation sources are the conductive loss due to the finite conductivity of the wall material ($\alpha_c$) and the loss caused by dielectric material ($\alpha_d$). In this sense, the attenuation constant is defined as $\alpha = \alpha_c + \alpha_d$\\

\noindent The conductor loss depends on the wall material and the geometry of the waveguide (width and height) whereas the dielectric loss mainly depends on the $tan (\delta)$ parameter of the dielectric material:

\begin{align}
    \text{Conductive loss:} \;\;\;\;\; \alpha_c & = \frac{R_s}{a^3 b \beta k \eta} (2b\pi^2 + a^3k^2) \\
    \text{Dielectric loss:} \;\;\;\;\;\alpha_d & = \frac{k^2 tan (\delta)}{2 \beta}
\end{align} 

\noindent where $k = \omega \sqrt{\mu \epsilon}$ is the wave number and $\eta = \sqrt{\frac{\mu}{\epsilon}}$ is the characteristic impedance of the dielectric material. The wall surface resistance ($R_s$) can be calculated as follows:

\begin{equation}
R_s = \sqrt{\frac{\omega \mu_0}{2 \sigma}}
\label{eq:WG_wallRs}
\end{equation}

\noindent where $\sigma$ is the conductivity of the walls.\\

\noindent The phase constant of the $TE_{10}$ mode is given by:

\begin{equation}
\beta  = \sqrt{k^2 - \left( \frac{\pi}{a}\right)^2}
\end{equation}

\noindent and the wave impedance of the TE modes can be calculated as:

\begin{equation}
Z_0 = \frac{k \cdot \eta}{\beta}
\end{equation}

\noindent Finally, the cutoff frequencies of the $TE_{10}$ and $TM_{11}$ modes are given by:

\begin{align}
f_c^{TE_{10}} & = \frac{1}{2 \pi \sqrt{\mu \epsilon}} \sqrt{\left( \frac{\pi}{a}\right)^2 } \\
f_c^{TM_{11}} &= \frac{1}{2 \pi \sqrt{\mu \epsilon}} \sqrt{\left( \frac{\pi}{a}\right)^2 +\left( \frac{\pi}{b}\right)^2 }
\end{align}


\noindent \textit{Reference}: \cite{Pozar}, pages 110-121

\section{Circular waveguide}

As seen in \ref{sec:rectWG}, most of practical applications involving waveguides are designed to propagate the dominant mode only. For this reason, the Qucs circular waveguide component only models the dominant mode ($TE_{11}$). Again, please notice that the simulation data is valid only for those frequencies between the $TE_{11}$ and the $TM_{01}$ cutoff, where only $TE_{11}$ propagation is guaranteed.

\begin{figure}[H]
\begin{center}
\psfrag{er}{$\mathrm{\varepsilon_r}$}
\includegraphics[width=0.5\linewidth]{circularwaveguide}
\end{center}
\caption{Circular waveguide of radius \textit{a} and length \textit{L}}
\label{fig:circwg}
\end{figure}

\noindent As introduced in \ref{sec:rectWG} the Qucs model extracts the quadripole parameters using the ideal transmission line model. The model parameters are obtained as follows:\\

\noindent The attenuation constant $\alpha = \alpha_c + \alpha_d$ involves both the loss in the conductive walls ($\alpha_c$) and the dielectric loss ($\alpha_d$):

\begin{align}
    \text{Conductive loss:} \;\;\;\;\; \alpha_c & = \frac{R_s}{a k \eta \beta} \left( k_c^2 + \frac{k^2}{2.389}\right) \\
    \text{Dielectric loss:} \;\;\;\;\;\alpha_d & = \frac{k^2 \cdot tan (\delta)}{ 2\beta}
\end{align}

\noindent where $k = \omega \sqrt{\mu \epsilon}$ is the wave number, $k_c = \frac{1.841}{a}$, $tan (\delta)$ is a parameter that accounts for the loss of the dielectric material and surface resistance of the wall ($R_s$) is calculated according to \ref{eq:WG_wallRs}.\\


\noindent The phase constant of the $TE_{11}$ mode is given by:

\begin{equation}
\beta = \sqrt{k^2 - \left( \frac{1.841}{a}\right)^2}
\end{equation}

\noindent As in the rectangular waveguide case, the characteristic impedance of a circular waveguide (TE modes) is given by:

\begin{equation}
Z_0 = \frac{k \eta}{\beta}
\end{equation}

\noindent Finally, the cutoff frequencies of the $TE_{11}$ and $TM_{01}$ modes are:

\begin{align}
f_c^{TE_{11}} & = \frac{1.841}{2 \pi a \sqrt{\mu \epsilon}} \\
f_c^{TM_{01}} & = \frac{2.405}{2 \pi a \sqrt{\mu \epsilon}}
\end{align}

\noindent \textit{Reference}: \cite{Pozar}, pages 121-130


\section{Twisted pair}
%\addcontentsline{toc}{section}{Twisted pair}

The twisted pair configurations as shown in fig. \ref{fig:twisted}
provides good low frequency shielding.  Undesired signals tend to be
coupled equally into eachline of the pair.  A differential receiver
will therefore completely cancel the interference.

\begin{figure}[ht]
\begin{center}
\psfrag{er}{$\mathrm{\varepsilon_r}$}
\includegraphics[width=0.3\linewidth]{twisted}
\end{center}
\caption{twisted pair configuration}
\label{fig:twisted}
\end{figure}
\FloatBarrier

\subsection{Quasi-static model}

According to P. Lefferson \cite{Lefferson} the characteristic
impedance and effective dielectric constant of a twisted pair can be
calculated as follows.
\begin{align}
Z_L &= \dfrac{Z_{F0}}{\pi\cdot\sqrt{\varepsilon_{r,eff}}}\cdot\textrm{acosh}\left(\dfrac{D}{d}\right)\\
\label{eq:TPereff}
\varepsilon_{r,eff} &= \varepsilon_{r,1} + q\cdot\left(\varepsilon_r - \varepsilon_{r,1}\right)
\end{align}

with
\begin{equation}
\label{eq:TPq}
q = 0.25 + 0.0004\cdot \theta^2
\;\;\;\; \textrm{ and } \;\;\;\;
\theta = \textrm{atan}\left(T\cdot\pi\cdot D\right)
\end{equation}

whereas $\theta$ is the pitch angle of the twist; the angle between
the twisted pair's center line and the twist.  It was found to be
optimal for $\theta$ to be between 20\degree and 45\degree.  $T$
denotes the twists per length.  Eq. \eqref{eq:TPq} is valid for film
insulations, for the softer PTFE material it should be modified as
follows.
\begin{equation}
q = 0.25 + 0.001\cdot \theta^2
\end{equation}

Assuming air as dielectric around the wires yields 1's replacing
$\varepsilon_{r,1}$ in eq. \eqref{eq:TPereff}.  The wire's total
length before twisting in terms of the number of turns $N$ is
\begin{equation}
l = N\cdot\pi\cdot D\cdot\sqrt{1 + \dfrac{1}{\tan^2{\theta}}}
\end{equation}

\subsection{Transmission losses}

The propagation constant $\gamma$ of a general transmission line is
given by
\begin{equation}
\gamma = \sqrt{\left(R' + j\omega L'\right)\cdot \left(G' + j\omega C'\right)}
\end{equation}

Using some transformations of the formula gives an expression with and
without the angular frequency.
\begin{equation}
\label{eq:gamma_rlgc}
\begin{split}
\gamma &= \sqrt{\left(R' + j\omega L'\right)\cdot \left(G' + j\omega C'\right)}\\
&= \sqrt{L'C'}\cdot\sqrt{\dfrac{R'G'}{L'C'} + j\omega\left(\dfrac{R'}{L'} + \dfrac{G'}{C'}\right) - \omega^2}\\
&= \sqrt{L'C'}\cdot\sqrt{\left(\dfrac{1}{2}\cdot\left(\dfrac{R'}{L'} + \dfrac{G'}{C'}\right) + j\omega\right)^2 - \dfrac{1}{4}\cdot\left(\dfrac{R'}{L'} + \dfrac{G'}{C'}\right)^2 + \dfrac{R'G'}{L'C'}}
\end{split}
\end{equation}

For high frequencies eq.{\eqref{eq:gamma_rlgc}} can be approximated to
\begin{equation}
\gamma \approx \sqrt{L'C'}\cdot\left(\dfrac{1}{2}\cdot\left(\dfrac{R'}{L'} + \dfrac{G'}{C'}\right) + j\omega\right)
\end{equation}

Thus the real part of the propagation constant $\gamma$ yields
\begin{equation}
\label{eq:alpha_freq}
\alpha = Re\left\{\gamma\right\} = \sqrt{L'C'}\cdot\dfrac{1}{2}\cdot\left(\dfrac{R'}{L'} + \dfrac{G'}{C'}\right)
\end{equation}

With
\begin{equation}
Z_L = \sqrt{\dfrac{L'}{C'}}
\end{equation}

the expression in eq.\eqref{eq:alpha_freq} can be written as
\begin{equation}
\alpha = \alpha_c + \alpha_d = \dfrac{1}{2}\cdot\left(\dfrac{R'}{Z_L} + G'Z_L\right)
\end{equation}

whereas $\alpha_c$ denotes the conductor losses and $\alpha_d$ the
dielectric losses.

\subsubsection{Conductor losses}

The sheet resistance R' of a transmission line conductor is given by
\begin{equation}
R' = \dfrac{\rho}{A_{eff}}
\end{equation}

whereas $\rho$ is the specific resistance of the conductor material
and $A_{eff}$ the effective area of the conductor perpendicular to the
propagation direction.  At higher frequencies the area of the
conductor is reduced by the skin effect.  The skin depth is given by
\begin{equation}
\delta_s = \sqrt{\dfrac{\rho}{\pi\cdot f\cdot\mu}}
\end{equation}

Thus the effective area of a single round wire yields
\begin{equation}
A_{eff} = \pi\cdot\left(r^2 - (r-\delta_s)^2\right)
\end{equation}

whereas $r$ denotes the radius of the wire.  This means the overall
conductor attenuation constant $\alpha_c$ for a single wire gives
\begin{equation}
\alpha_c = \dfrac{R'}{2\cdot Z_L} = \dfrac{\rho}{2\cdot Z_L\cdot\pi\cdot\left(r^2 - (r-\delta_s)^2\right)}
\end{equation}

\subsubsection{Dielectric losses}

The dielectric losses are determined by the dielectric loss tangent.
\begin{equation}
\label{eq:g_dash}
\tan{ \delta_d } = \dfrac{G'}{\omega C'}
\;\;\;\; \rightarrow \;\;\;\;
G' = \omega C' \cdot \tan{ \delta_d }
\end{equation}

With
\begin{equation}
C' = \dfrac{1}{\omega}\cdot Im \left\{\dfrac{\gamma}{Z_L}\right\}
\end{equation}

the equation \eqref{eq:g_dash} can be rewritten to
\begin{equation}
\begin{split}
G' &= \dfrac{\beta}{Z_L}\cdot \tan{ \delta_d } = \dfrac{\omega}{v_{ph}\cdot Z_L}\cdot \tan{ \delta_d } \\
&= \dfrac{2\pi\cdot f\cdot \sqrt{\varepsilon_{r,eff}}}{c_0\cdot Z_L}\cdot \tan{ \delta_d } = \dfrac{2\pi\cdot \sqrt{\varepsilon_{r,eff}}}{\lambda_0\cdot Z_L}\cdot \tan{ \delta_d }
\end{split}
\end{equation}

whereas $v_{ph}$ denotes the phase velocity, $c_0$ the speed of light,
$\varepsilon_{r,eff}$ the effective dielectric constant and
$\lambda_0$ the freespace wavelength.  With these expressions at hand
it is possible to find a formula for the dielectric losses of the
transmission line.
\begin{equation}
\alpha_d = \dfrac{1}{2}\cdot G'Z_L = \dfrac{\pi\cdot \sqrt{\varepsilon_{r,eff}}}{\lambda_0}\cdot \tan{ \delta_d }
\end{equation}

\subsubsection{Overall losses of the twisted pair configuration}

Transmission losses consist of conductor losses, dielectric losses as
well as radiation losses.  The above expressions for the conductor and
dielectric losses are considered to be first order approximations.
The conductor losses have been derived for a single round wire.  The
overall conductor losses due to the twin wires must be doubled.  The
dielectric losses can be used as is.  Radiation losses are neglected.
